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❈♦♥s✐st❡♥t ♠♦❞❡❧ s❡❧❡❝t✐♦♥ ❝r✐t❡r✐❛ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ❢♦r ❝♦♠♠♦♥ t✐♠❡ s❡r✐❡s ♠♦❞❡❧s

❏♦✐♥t ✇♦r❦ ✇✐t❤ ❑✳ ❑❛r❡ ✭P❛r✐s ✶✮ ❛♥❞ ❲✳ ❑❡♥❣♥❡ ✭❈❡r❣②✮ ❏❡❛♥✲▼❛r❝ ❇❛r❞❡t✱ ❙❆▼▼✱ ❯♥✐✈❡rs✐té P❛r✐s ✶✱ ❋r❛♥❝❡ ❜❛r❞❡t❅✉♥✐✈✲♣❛r✐s✶✳❢r ▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡ ✷ ❏✉♥❡ ✷✵✷✵

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶ ✴ ✸✷

✶ ❆♥ ❡①❛♠♣❧❡ ✷ ❈❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸ ●❛✉ss✐❛♥ ◗✉❛s✐✲▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❊st✐♠❛t✐♦♥ ✹ ❈♦♥s✐st❡♥❝② ♦❢ ❛ ♣❡♥❛❧✐③❡❞ ◗▼▲ ❝r✐t❡r✐♦♥ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ✺ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❖✉t❧✐♥❡

✶ ❆♥ ❡①❛♠♣❧❡ ✷ ❈❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸ ●❛✉ss✐❛♥ ◗✉❛s✐✲▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❊st✐♠❛t✐♦♥ ✹ ❈♦♥s✐st❡♥❝② ♦❢ ❛ ♣❡♥❛❧✐③❡❞ ◗▼▲ ❝r✐t❡r✐♦♥ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ✺ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✷ ✴ ✸✷

❊①❛♠♣❧❡

❖❜s❡r✈❡ t❤❡ tr❛❥❡❝t♦r② ♦❢ t❤❡ ❧♦❣❛r✐t❤♠✐❝ r❡t✉r♥s ♦❢ ❙✫P ✺✵✵ ✿

2012 2013 2014 2015 2016 2017 −0.04 −0.02 0.00 0.02 0.04 Date logSP500

= ⇒ ❚✇♦ ❛✐♠s ✿ ❈❤♦s❡ ❛♥ ✧♦♣t✐♠❛❧✧ ♠♦❞❡❧ ❢♦r t❤❡s❡ ❞❛t❛ ❀ ❚❡st ✐ts ❣♦♦❞♥❡ss✲♦❢✲✜t✳

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸ ✴ ✸✷

❚✇♦ ✐♥t✉✐t✐✈❡ ❞❡✜♥✐t✐♦♥s

▲❡t (Xt)t∈Z ❜❡ ❛ t✐♠❡ s❡r✐❡s ✭s❡q✉❡♥❝❡ ♦❢ r✳✈✳ ♦♥ (Ω, A, I P)✮ (Xt)t∈Z ✐s ❛ st❛t✐♦♥❛r② ♣r♦❝❡ss ✐❢ ∀k ∈ I N∗✱ ∀(t✶, . . . , tk) ∈ Zk✱

• Xt✶, . . . , Xtk

L ∼

• Xt✶+h, . . . , Xtk+h
• ❢♦r ❛❧❧ h ∈ Z✳

❆ss✉♠❡ t❤❛t (ξt)t∈Z ✐s ❛ ✇❤✐t❡ ♥♦✐s❡ ✭❝❡♥t❡r❡❞ ✐✳✐✳❞✳r✳✈✳✮ (Xt)t∈Z ❝❛✉s❛❧ ♣r♦❝❡ss ✐❢ ∃H : I RI

N → I

R s✉❝❤ ❛s Xt = H

• (ξt−k)k≥✵
• ✳

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✹ ✴ ✸✷

❆❘▼❆ ♣r♦❝❡ss❡s ✭❲❤✐tt❧❡✱ ✶✾✺✶✮

❲✐t❤ (ξt)t∈Z ❛ ✇❤✐t❡ ♥♦✐s❡✱ (ai) ∈ I Rp✱ (bj) ∈ I Rq ❆❘▼❆✭p, q✮ ♣r♦❝❡ss ✿ ✇✐t❤ ap = ✵ ❛♥❞ bq = ✵✱ ❢♦r ❛♥② t ∈ Z Xt + a✶Xt−✶ + · · · + apXt−p = ξt + b✶ξt−✶ + · · · + bqξt−q ❙t❛t✐♦♥❛r✐t② ❛♥❞ ❝❛✉s❛❧✐t② ✿ ✶ + a✶z + · · · + apzp = ✵ ❢♦r ❛♥② |z| ≤ ✶✳

200 400 600 800 1000 −15 −10 −5 5 10

❆❘▼❆✭✶, ✶✮

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✺ ✴ ✸✷

• ❆❘❈❍ ♣r♦❝❡ss❡s ✭❊♥❣❡❧✱ ✶✾✽✷✮ ✭❇♦❧❧❡rs❡✈✱ ✶✾✽✻✮

❲✐t❤ (ξt)t∈Z ❛ ✇❤✐t❡ ♥♦✐s❡✱ (ci) ∈ I Rp

+✱ (dj) ∈ I

Rq

+

• ❆❘❈❍✭p, q✮ ♣r♦❝❡ss ✿ ✇✐t❤ c✵, cp > ✵ ❛♥❞ dq > ✵✱ ❢♦r ❛♥② t ∈ Z

Xt = σt ξt, σ✷

t

= c✵ + c✶X ✷

t−✶ + · · · + cpX ✷ t−p + d✶σ✷ t−✶ + · · · + dqσ✷ t−q

q

j=✶ dj + I

E(ξ✷

✵) p i=✶ ci < ✶ =

⇒ ❙t❛t✐♦♥❛r✐t② ❛♥❞ ❝❛✉s❛❧✐t②

200 400 600 800 1000 −30 −20 −10 10 20 30

• ❆❘❈❍✭✶, ✶✮

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✻ ✴ ✸✷

▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❛♥❞ ●♦♦❞♥❡ss✲♦❢✲✜t t❡st

❈♦♥s✐❞❡r ❛ ❢❛♠✐❧② M ♦❢ ♠♦❞❡❧s✳ ❋♦r ✐♥st❛♥❝❡✱ M =

• ❆❘▼❆(p, q) ♦r ●❆❘❈❍(p′, q′),

✇✐t❤ ✵ ≤ p, p′ ≤ pmax, ✵ ≤ q, q′ ≤ qmax

• ❲❡ ✇❛♥t t♦ ✿

❈❤♦s❡ ❛♥ ✧♦♣t✐♠❛❧✧ ♠♦❞❡❧ ✐♥ M ❢♦r (X✶, . . . , Xn) ❀ ❊st✐♠❛t❡ ✐ts ♣❛r❛♠❡t❡rs ❀ ❚❡st ✐ts ❣♦♦❞♥❡ss✲♦❢✲✜t✳

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✼ ✴ ✸✷

❖✉t❧✐♥❡

✶ ❆♥ ❡①❛♠♣❧❡ ✷ ❈❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸ ●❛✉ss✐❛♥ ◗✉❛s✐✲▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❊st✐♠❛t✐♦♥ ✹ ❈♦♥s✐st❡♥❝② ♦❢ ❛ ♣❡♥❛❧✐③❡❞ ◗▼▲ ❝r✐t❡r✐♦♥ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ✺ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✽ ✴ ✸✷

❊①❛♠♣❧❡s ✿ ❈❛✉s❛❧ ❆❘❬∞❪ ❛♥❞ ❆❘❈❍✭∞✮ ♠♦❞❡❧s

❲✐t❤ (ξt)t∈Z ❛ ✇❤✐t❡ ♥♦✐s❡✱ ❆❘(∞) ♣r♦❝❡ss❡s Xt =

∞

• i=✶

θiXt−i + ξt = ⇒ ❈❛✉s❛❧ ❆❘▼❆✭p, q✮ ♣r♦❝❡ss❡s Xt +

p

• i=✶

aiXt−i = ξt +

q

• i=✶

biξt−i✳ ❆❘❈❍✭∞✮ ♣r♦❝❡ss❡s✱ ✭❘♦❜✐♥s♦♥✱ ✶✾✾✶✮✱ ✇✐t❤ b✵ > ✵ ❛♥❞ bj ≥ ✵ Xt = σtξt, σ✷

t

= φ✵ + ∞

j=✶ φjX ✷ t−j.

= ⇒ ●❆❘❈❍✭p, q✮ ♣r♦❝❡ss❡s✱ ✇✐t❤ c✵ > ✵✱ cj, dj ≥ ✵✱ cp, dq > ✵ Xt = σtξt, σ✷

t

= c✵ + p

j=✶ cjX ✷ t−j + q j=✶ djσ✷ t−j

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✾ ✴ ✸✷

❆ ❝♦♠♠♦♥ ❢r❛♠❡ ❢♦r st✉❞②✐♥❣ t✐♠❡ s❡r✐❡s

❆ ❝♦♠♠♦♥ ❝❧❛ss ♦❢ ♠♦❞❡❧s ❢♦r ❆❘✱ ❆❘▼❆✱ ❆❘❈❍ ❛♥❞ ●❆❘❈❍ ♣r♦❝❡ss❡s ✿

❈❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✿ ❝❧❛ss CA(M, f )

Xt = M(Xt−✶, Xt−✷, . . .) ξt + f (Xt−✶, Xt−✷, . . .), ∀ t ∈ Z, ❛✳s✳. M(·) ❛♥❞ f (·) ❛r❡ r❡❛❧ ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ♦♥ I RI

N ❀

(ξt)t∈Z ❛ ✇❤✐t❡ ♥♦✐s❡ ✇✐t❤ I E(ξ✵) = ✵ ❛♥❞ I E

• |ξ✵|r

< ∞✱ r ≥ ✶✳

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶✵ ✴ ✸✷

❊①t❡♥s✐♦♥s ♦❢ ✉♥✐✈❛r✐❛t❡ ❆❘❈❍ ♠♦❞❡❧s

❚●❆❘❈❍✭∞✮ ♣r♦❝❡ss❡s✱ ✭❩❛❦♦ï❛♥✱ ✶✾✾✹✮✱ ✇✐t❤ b✵, b+

j , b− j ≥ ✵

   Xt = σt ξt, σt = b✵ +

∞

• j=✶
• b+

j max(Xt−j, ✵) − b− j min(Xt−j, ✵)

✳ ❆P❆❘❈❍✭δ, p, q✮ ♣r♦❝❡ss❡s✱ ✭❉✐♥❣ ❡t ❛❧✳✱ ✶✾✾✸✮    Xt = σt ζt, σδ

t

= ω +

p

• j=✶

αi(|Xt−i| − γiXt−i)δ + q

j=✶ βjσδ t−j,

✇✐t❤ δ ≥ ✶✱ ω > ✵✱ −✶ < γi < ✶ ❛♥❞ αi, βj ≥ ✵✳

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶✶ ✴ ✸✷

❈♦♠❜✐♥❛t✐♦♥s ♦❢ ♠♦❞❡❧s

❆❘▼❆✲●❆❘❈❍ ♣r♦❝❡ss❡s✱ ✭❉✐♥❣ ❡t ❛❧✳✱ ✶✾✾✸✱ ▲✐♥❣ ❛♥❞ ▼❝❆❧❡❡r✱ ✷✵✵✸✮          Xt =

p

• i=✶

aiXt−i + εt +

q

• j=✶

bjεt−j, εt = σt ζt, ✇✐t❤ σ✷

t = c✵ + p′

• i=✶

ciε✷

t−i + q′

• j=✶

djσ✷

t−j

❆❘▼❆✲❆P❆❘❈❍ ♣r♦❝❡ss❡s✱ ✭❉✐♥❣ ❡t ❛❧✳✱ ✶✾✾✸✮          Xt =

p

• i=✶

aiXt−i + εt +

q

• j=✶

bjεt−j, εt = σt ζt, ✇✐t❤ σδ

t = ω + p′

• j=✶

αi(|Xt−i| − γiXt−i)δ +

q′

• j=✶

βjσδ

t−j

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶✷ ✴ ✸✷

❊①✐st❡♥❝❡ ❛♥❞ st❛t✐♦♥❛r✐t② ♦❢ ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s

Xt = M(Xt−✶, Xt−✷, . . .) ξt + f (Xt−✶, Xt−✷, . . .), ∀ t ∈ Z, ❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t f ❛♥❞ M s❛t✐s❢② ▲✐♣s❝❤✐t③✐❛♥ ❝♦♥❞✐t✐♦♥s ✿ |f (x) − f (y)| ≤ ∞

j=✶ αj(f )|xj − yj|

|M(x) − M(y)| ≤ ∞

j=✶ αj(M)|xj − yj|.

❢♦r x = (xj)j∈I

N ❛♥❞ y = (yj)j∈I N t✇♦ s❡q✉❡♥❝❡s ♦❢ I

R∞✳

Pr♦♣♦s✐t✐♦♥ ✭❢r♦♠ ❉♦✉❦❤❛♥ ❛♥❞ ❲✐♥t❡♥❜❡r❣❡r✱ ✷✵✵✼✮

■❢ ∞

j=✶ αj(f ) +

• I

E(|ξ✵|r) ✶/r ∞

j=✶ αj(M) < ✶✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❝❛✉s❛❧

s♦❧✉t✐♦♥ (Xt)t∈Z ✇❤✐❝❤ ✐s st❛t✐♦♥❛r②✱ ❡r❣♦❞✐❝✱ s✉❝❤ ❛s I E(|X✵|r) < ∞✳

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶✸ ✴ ✸✷

❊①❛♠♣❧❡s

❈♦♥❞✐t✐♦♥s ♦♥ st❛t✐♦♥❛r✐t② ❜❡❝♦♠❡ ✿ ❈❛✉s❛❧ ❆❘[∞] ✿ Xt = ∞

j=✵ ajξt−j =

⇒ ∞

j=✵ |aj| < ✶ ❀

❈❛✉s❛❧ ❆❘❈❍[∞] ✿ Xt = ξt

• c✵ + ∞

j=✶ cjX ✷ t−j =

⇒

• I

E

• |ξ✵|r✶/r ∞

j=✶ cj < ✶ ❀

❈❛✉s❛❧ ❚❆❘❈❍[∞] ✿ Xt = ξt

• b✵ + ∞

j=✶

• b+

j max(Xt−j, ✵) − b− j min(Xt−j, ✵)

• =

⇒

• I

E

• |ξ✵|r✶/r ∞

j=✶ max

• b−

j , b+ j

• < ✶ ❀

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶✹ ✴ ✸✷

❖✉t❧✐♥❡

✶ ❆♥ ❡①❛♠♣❧❡ ✷ ❈❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸ ●❛✉ss✐❛♥ ◗✉❛s✐✲▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❊st✐♠❛t✐♦♥ ✹ ❈♦♥s✐st❡♥❝② ♦❢ ❛ ♣❡♥❛❧✐③❡❞ ◗▼▲ ❝r✐t❡r✐♦♥ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ✺ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶✺ ✴ ✸✷

• ❛✉ss✐❛♥ ◗▼▲❊ ♦❢ ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧

▲❡t (X✶, . . . , Xn) ❛♥ ♦❜s❡r✈❡❞ tr❛❥❡❝t♦r② ♦❢ ❛♥ CA(Mθ∗, fθ∗) Xt = Mθ∗(Xt−✶, Xt−✷, . . .) ξt + fθ∗(Xt−✶, Xt−✷, . . .), ∀ t ∈ Z ❲✐t❤ f t

θ = fθ(Xt−✶, Xt−✷, . . .)✱ Mt θ = Mθ(Xt−✶, Xt−✷, . . .)✱

• ❛✉ss✐❛♥ ❝♦♥❞✐t✐♦♥❛❧ ❧♦❣✲❞❡♥s✐t② ✿ qt(θ) = −✶

✷ (Xt − f t

θ )✷

(Mt

θ)✷

+ log(Mt

θ)✷

▲❡t f t

θ = fθ(Xt−✶, . . . , X✶, ✵, · · · ) ❛♥❞

Mt

θ = Mθ(Xt−✶, . . . , X✶, ✵, · · · )

• qt(θ) = −✶

✷ (Xt − f t

θ )✷

( Mt

θ)✷

+ log

• (

Mt

θ)✷

. = ⇒ ●❛✉ss✐❛♥ ◗▼▲❊ ✿ θn = ❛r❣max

θ∈Θ

• Ln(θ) ✇✐t❤

Ln(θ) =

n

• t=✶
• qt(θ)✳

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶✻ ✴ ✸✷

❆ss✉♠♣t✐♦♥s ❛♥❞ str♦♥❣ ❝♦♥s✐st❡♥❝②

❲❡ ❛ss✉♠❡ ✿ ❈✵ ✿ r ≥ ✷ ❛♥❞ I E(ξ✷

✵) = ✶ ❀

❈✶ ✿ Θ ✐s ❛ ❝♦♠♣❛❝t s❡t ✐♥❝❧✉❞❡❞ ✐♥ Θ(r)=

• θ ∈ I

Rd

∞

• j=✶

α(✵)

j

(fθ) + (I E(|ξ✵|r))✶/r

∞

• j=✶

α(✵)

j

(Mθ) < ✶

• .

❈✷ ✿ ∃M > ✵ s✉❝❤ t❤❛t Mθ(x) ≥ M ❢♦r ❛❧❧ θ ∈ Θ✱ x ∈ I RI

N✳

❈✸ ✿ Mθ ❛♥❞ fθ ❛r❡ s✉❝❤ t❤❛t ❢♦r ❛❧❧ θ✶, θ✷ ∈ Θ✱ t❤❡♥ ✿

• Mθ✶ = Mθ✷

❛♥❞ fθ✶ = fθ✷

• =

⇒ θ✶ = θ✷ ❆✭Kθ, Θ✮ ✿ ❚❤❡r❡ ❡①✐sts

• αj(Kθ, Θ)
• j s✉❝❤ t❤❛t ∀x✱ y ∈ I

R∞ sup

θ∈Θ

• Kθ(x) − Kθ(y)
• ≤

∞

• j=✶

αj(Kθ, Θ)|xj − yj|, ✇✐t❤

∞

• j=✶

αj(Kθ, Θ) < ∞.

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶✼ ✴ ✸✷

❙tr♦♥❣ ❝♦♥s✐st❡♥❝②

❚❤é♦rè♠❡ ✭❇❛r❞❡t ❛♥❞ ❲✐♥t❡♥❜❡r❣❡r✱ ✷✵✵✾✮

❆ss✉♠❡ r ≥ ✷✱ Θ ⊂ Θ(✷)✱ ❈♦♥❞✐t✐♦♥s ❈✵✲✸ ❛♥❞ ❆✭fθ, Θ✮ ❛♥❞ ❆✭Mθ, Θ✮ ✇✐t❤ αj(fθ, Θ) + αj(Mθ, Θ) = O

• j−ℓ

❢♦r s♦♠❡ ℓ > min(✶, ✸/r). ❚❤❡♥ t❤❡ ◗▼▲❊ θn ✐s str♦♥❣❧② ❝♦♥s✐st❡♥t✱ ✐✳❡✳ θn

a.s.

− →

n→∞ θ∗.

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶✽ ✴ ✸✷

❆s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t②

❚❤é♦rè♠❡ ✭❇❛r❞❡t ❛♥❞ ❲✐♥t❡♥❜❡r❣❡r✱ ✷✵✵✾✮

❯♥❞❡r ❝♦♥❞✐t✐♦♥s ♦❢ ❙▲▲◆✱ ❛♥❞ ✐❢ r ≥ ✹✱ ✐❢ θ∗ ∈

• Θ ∩Θ(✹) ❛♥❞ ✐❢ ❆✭Kθ, Θ✮✱

❆✭∂θKθ, Θ✮ ❛♥❞ ❆✭∂✷

θ✷Kθ, Θ✮ ❤♦❧❞ ❢♦r Kθ = fθ ♦r Mθ✱ ❛♥❞ ✐❢

αj(∂θfθ, Θ) + αj(∂θMθ, Θ) = O

• j−ℓ′

❢♦r s♦♠❡ ℓ′ > ✶, ✭✶✮ t❤❡♥ t❤❡ ◗▼▲❊ θn ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧✱ ✐✳❡✳✱ t❤❡r❡ ❡①✐sts ♠❛tr✐① F(θ∗)−✶ ❛♥❞ G(θ∗) s✉❝❤ t❤❛t √n

• θn − θ∗

L

− →

n→∞ Nd

• ✵ , F(θ∗)−✶G(θ∗)F(θ∗)−✶

. ✭✷✮ ❈♦✉❧❞ ❜❡ ❛♣♣❧✐❡❞ t♦ ❛❧❧ ❝✐t❡❞ ♣r♦❝❡ss❡s ❆❘▼❆✱ ❆❘❈❍✱ ❆P❆❘❈❍✱✳✳✳ ❇✉t r❡q✉✐r❡s r ≥ ✹ ❛♥❞ ♥♦t ✈❡r② r♦❜✉st✳

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶✾ ✴ ✸✷

❖✉t❧✐♥❡

✶ ❆♥ ❡①❛♠♣❧❡ ✷ ❈❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸ ●❛✉ss✐❛♥ ◗✉❛s✐✲▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❊st✐♠❛t✐♦♥ ✹ ❈♦♥s✐st❡♥❝② ♦❢ ❛ ♣❡♥❛❧✐③❡❞ ◗▼▲ ❝r✐t❡r✐♦♥ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ✺ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✷✵ ✴ ✸✷

❆❞❞✐t✐✈✐t② ♦❢ ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s

Pr♦♣♦s✐t✐♦♥

▲❡t Θ✶ ⊂ Rd✶✱ Θ✷ ⊂ Rd✶✱ M(✶)

θ✶ , f (✶) θ✶ , M(✷) θ✷ , f (✷) θ✷

❢♦r θ✶ ∈ Θ✶✱ θ✷ ∈ Θ✷✳ ❚❤❡r❡ ❡①✐st max(d✶, d✷) ≤ d ≤ d✶ + d✷✱ Θ ⊂ I Rd✱ ❛♥❞ Mθ, fθ ✇✐t❤ θ ∈ Θ✱ s✉❝❤ ❛s ❢♦r ❛♥② θ✶ ∈ Θ✶ ⊂ Rd✶ ❛♥❞ θ✷ ∈ Θ✷ ⊂ Rd✷✱

• CA
• M(✶)

θ✶ , f (✶) θ✶

CA

• M(✷)

θ✷ , f (✷) θ✷

• ⊂
• CA
• Mθ, fθ
• .

❈♦♥s❡q✉❡♥❝❡ ✿ ❋♦r ❛♥② ❢❛♠✐❧② M =

i∈I CA

• M(i)

θi , f (i) θi

• ✱

= ⇒ M =

• i∈I
• CA
• Mθ, fθ
• θ∈Θi⊂I

Rd

M ❢❛♠✐❧② ♦❢ CA ♠♦❞❡❧s ⇔ M ∼ {m ⊂ {✶, . . . , d}

• ✱

θ ∈ Θ(m) ⊂

• (x✶, . . . , xd) ∈ I

Rd, xi = ✵ ✐❢ i / ∈ m

• ❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮

▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✷✶ ✴ ✸✷

P❡♥❛❧✐③❡❞ ◗✉❛s✐✲▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❝r✐t❡r✐♦♥

▲❡t (X✶, . . . , Xn) ❛♥ ♦❜s❡r✈❡❞ tr❛❥❡❝t♦r②✳ ❋♦r m ∈ M✱ ❞❡✜♥❡ ✿     

• θ(m)

= argmax

θ∈Θ(m)

• Ln(θ)
• m

= argmin

m∈M

• C(m)

✇✐t❤

• C(m) = −✷

Ln

• θ(m)
• + |m| κn,

✉s✐♥❣ (κn)n ❛♥ ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ r❡❛❧ ♥✉♠❜❡rs ❀ |m| ❞❡♥♦t❡s t❤❡ ❝❛r❞✐♥❛❧ ♦❢ m✱ s✉❜s❡t ♦❢ {✶, . . . , d}✳

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✷✷ ✴ ✸✷

❈♦♥s✐st❡♥❝②

❚❤é♦rè♠❡

▲❡t (X✶, . . . , Xn) ❜❡ ❛♥ ♦❜s❡r✈❡❞ tr❛❥❡❝t♦r② ♦❢ CA(Mθ∗, fθ∗) ✇❤❡r❡ θ∗ ✉♥❦♥♦✇♥ ✐♥ Θ ⊂ Θ(r) ⊂ I Rd ✇✐t❤ r ≥ ✹✳ ❯♥❞❡r ♣r❡✈✐♦✉s ❛ss✉♠♣t✐♦♥s ❛♥❞ ✐❢

• k≥✶

✶ κk

• j≥k

αj(fθ, Θ) + αj(Mθ, Θ) + αj(∂θfθ, Θ) + αj(∂θMθ, Θ) < ∞, t❤❡♥ I P( m = m∗) − →

n→+∞ ✶.

❈♦♥s❡q✉❡♥❝❡ ✿ ■❢ αj(fθ, Θ) + αj(Mθ, Θ) + αj(∂θfθ, Θ) + αj(∂θMθ, Θ) = O(ρj)✱ |ρ| < ✶✱ κn → ∞ s✉✣❝✐❡♥t ✿ ❇■❈ ❢♦r ❆❘▼❆✱ ●❆❘❈❍✱ ❆P❆❘❈❍✱✳✳✳✱ ♣r♦❝❡ss❡s✳ ■❢ αj(fθ, Θ) + αj(Mθ, Θ) + αj(∂θfθ, Θ) + αj(∂θMθ, Θ) = O(j−γ)✱ γ > ✶✱ κn = O(nδ) ✇✐t❤ δ > ✷− γ ✿ ♥♦t ✈❛❧✐❞ ❢♦r ❇■❈ ❢♦r ❆❘(∞)✱ ❆❘❈❍(∞)✱✳✳✳

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✷✸ ✴ ✸✷

❊st✐♠❛t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs

❚❤é♦rè♠❡

❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ t❤❡ ♣r❡✈✐♦✉s ❚❤❡♦r❡♠✱ t❤❡♥ √n

• θn(

m)

• i − (θ∗)i
• i∈

m L

− →

n→∞ Nd

• ✵ , F(θ∗, m∗)−✶G(θ∗, m∗)F(θ∗, m∗)−✶

✭✸✮ ✇❤❡r❡ F ❛♥❞ G ❛r❡ ❞❡✜♥❡❞ ✐♥ ❈▲❚✳ = ⇒ ❙❛♠❡ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡ ✇✐t❤ ♦r ✇✐t❤♦✉t t❤❡ ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ♠♦❞❡❧

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✷✹ ✴ ✸✷

P♦rt♠❛♥t❡❛✉ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ✭✶✮

❉❡✜♥❡ ✿ ❘❡s✐❞✉❛❧s ✿ ξk = Xk − f t

• θ(

m)

• Mt
• θ(

m)

❈♦✈❛r✐♦❣r❛♠ ♦❢ sq✉❛r❡ r❡s✐❞✉❛❧s ✿ r(k) = ✶ n

n−k

• j=✶
• ξ✷

j

ξ✷

j+k − ✶ ❀

❈♦rr❡❧♦❣r❛♠ ♦❢ sq✉❛r❡ r❡s✐❞✉❛❧s ✿ ρ(k) = r(k)

• r(✵) ❀

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✷✺ ✴ ✸✷

P♦rt♠❛♥t❡❛✉ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ✭✷✮

❚❤é♦rè♠❡

❯♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ❛♥❞ ✐❢ I E(ξ✸

✵) = ✵ ✿

✶ ❲✐t❤ V (θ∗) ❛♥ ❡①♣❧✐❝✐t ❞❡✜♥✐t❡ ♣♦s✐t✐✈❡ ♠❛tr✐①✱ ✇❡ ❤❛✈❡ ✿

√n

• ρ(✶), . . . ,

ρ(K)

• L

− →

n→∞ NK

• ✵ , V (θ∗)
• .

✷ ■❢ ✇❡ ❞❡✜♥❡

QK = n t

• ρ(✶), . . . ,

ρ(K) V ( θ( m)) −✶

• ρ(✶), . . . ,

ρ(K)

• ✱

t❤❡♥

• QK

L

− →

n→∞ χ✷(K).

= ⇒ ❚❡st H✵ : X ∈ AC(Mθ∗, fθ∗) H✶ : X / ∈ AC(Mθ∗, fθ∗)

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✷✻ ✴ ✸✷

❖✉t❧✐♥❡

✶ ❆♥ ❡①❛♠♣❧❡ ✷ ❈❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸ ●❛✉ss✐❛♥ ◗✉❛s✐✲▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❊st✐♠❛t✐♦♥ ✹ ❈♦♥s✐st❡♥❝② ♦❢ ❛ ♣❡♥❛❧✐③❡❞ ◗▼▲ ❝r✐t❡r✐♦♥ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ✺ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✷✼ ✴ ✸✷

❙✐♠✉❧❛t✐♦♥ r❡s✉❧ts ❢♦r ❝❧❛ss✐❝❛❧ ♠♦❞❡❧s

✶ ▼♦❞❡❧ ✶✱ ❆❘✭✷✮ ✿ Xt = ✵.✹Xt−✶ + ✵.✹Xt−✷ + ξt ✷ ▼♦❞❡❧ ✷✱ ❆❘▼❆✭✶✱✶✮ ✿ Xt = ✵.✸Xt−✶ + ξt + ✵.✺ξt−✶ ✸ ▼♦❞❡❧ ✸✱ ❆❘❈❍✭✷✮ ✿ Xt = ξt

• ✵.✷ + ✵.✹X ✷

t−✶ + ✵.✷X ✷ t−✷

n ✶✵✵ ✺✵✵ ✶✵✵✵ ✷✵✵✵ log n √n

• κn

log n √n

• κn

log n √n

• κn

log n √n

• κn

❲ ✷✶.✹ ✸✷.✸ ✶✽.✹ ✶.✼ ✵.✽ ✵.✾ ✵.✽ ✵.✶ ✵.✶ ✵.✷ ✵ ✵ ▼✶ ❚ ✼✹.✷ ✻✼.✻ ✼✾.✼ ✾✼.✷ ✾✾.✷ ✾✾.✶ ✾✽.✷ ✾✾.✾ ✾✾.✾ ✾✾.✷ ✶✵✵ ✶✵✵ ❖ ✹.✹ ✵.✶ ✶.✾ ✶.✶ ✵ ✵ ✶.✵ ✵ ✵ ✵.✻ ✵ ✵ ❲ ✸✵.✹ ✺✼.✼ ✷✽.✵ ✹.✽ ✹.✷ ✹.✵ ✵.✼ ✵.✸ ✵.✸ ✵.✹ ✵ ✵ ▼✷ ❚ ✻✹.✶ ✹✷.✶ ✻✼.✸ ✾✸.✻ ✾✺.✽ ✾✺.✽ ✾✽.✷ ✾✾.✼ ✾✾.✻ ✾✾.✷ ✶✵✵ ✶✵✵ ❖ ✺.✺ ✵.✷ ✹.✼ ✶.✻ ✵ ✵.✷ ✶.✶ ✵ ✵.✶ ✵.✹ ✵ ✵ ❲ ✼✻.✶ ✾✵.✽ ✺✸.✺ ✷✼.✸ ✻✼.✶ ✶✽.✵ ✶✹.✵ ✹✶.✺ ✶✸.✸ ✹.✻ ✶✷.✵ ✹.✻ ▼✸ ❚ ✷✸.✽ ✾.✷ ✸✾.✽ ✼✷.✼ ✸✷.✾ ✼✾.✾ ✽✺.✾ ✺✽.✺ ✽✻.✼ ✾✺.✹ ✽✽.✵ ✾✺.✹ ❖ ✵.✶ ✵ ✻.✼ ✵ ✵ ✷.✶ ✵.✶ ✵ ✵ ✵ ✵ ✵ ❲ ✽✸.✽ ✾✹.✸ ✼✸.✹ ✷✷.✶ ✻✶.✺ ✷✵.✹ ✺.✽ ✸✶.✸ ✺.✼ ✶.✽ ✻.✷ ✵.✼ ▼✹ ❚ ✶✺.✾ ✺.✼ ✷✶.✻ ✼✼.✺ ✸✽.✺ ✼✺.✾ ✾✸.✷ ✻✽.✼ ✾✷.✻ ✾✽.✵ ✾✸.✽ ✾✾.✸ ❖ ✵ ✸ ✵ ✺ ✵ ✵ ✹ ✵ ✸ ✼ ✶ ✵ ✵ ✶ ✼ ✵ ✷ ✵ ✵

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✷✽ ✴ ✸✷

❙✐♠✉❧❛t✐♦♥ r❡s✉❧ts ❢♦r ❝❧❛ss✐❝❛❧ ♠♦❞❡❧s

✶ ▼♦❞❡❧ ✶✱ ❆❘✭✷✮ ✿ Xt = ✵.✹Xt−✶ + ✵.✹Xt−✷ + ξt ✷ ▼♦❞❡❧ ✷✱ ❆❘▼❆✭✶✱✶✮ ✿ Xt = ✵.✸Xt−✶ + ξt + ✵.✺ξt−✶ ✸ ▼♦❞❡❧ ✸✱ ❆❘❈❍✭✷✮ ✿ Xt = ξt

• ✵.✷ + ✵.✹X ✷

t−✶ + ✵.✷X ✷ t−✷

♥ ✶✵✵ ✺✵✵ ✶✵✵✵ ✷✵✵✵ s✐③❡ ♣♦✇❡r s✐③❡ ♣♦✇❡r s✐③❡ ♣♦✇❡r s✐③❡ ♣♦✇❡r K = ✸ ▼♦❞❡❧ ✶ ✸✳✸ ✶✵✳✾ ✻✳✷ ✺✷✳✷ ✸✳✺ ✽✹✳✽ ✺✳✵ ✾✽✳✷ ▼♦❞❡❧ ✷ ✸✳✸ ✼✳✵ ✹✳✽ ✷✸✳✸ ✻✳✷ ✹✷✳✹ ✹✳✾ ✼✵✳✹ ▼♦❞❡❧ ✸ ✹✳✻ ✻✳✹ ✽✳✹ ✹✹✳✶ ✶✹✳✸ ✽✶✳✵ ✸✻✳✾ ✾✾✳✹ K = ✻ ▼♦❞❡❧ ✶ ✷✳✾ ✾✳✶ ✹✳✾ ✹✷✳✵ ✹✳✹ ✼✻✳✸ ✹✳✺ ✾✼✳✻ ▼♦❞❡❧ ✷ ✸✳✵ ✻✳✸ ✺✳✷ ✶✽✳✵ ✺✳✶ ✸✺✳✶ ✹✳✻ ✻✵✳✷ ▼♦❞❡❧ ✸ ✹✳✺ ✶✷✳✻ ✶✶✳✶ ✻✹✳✹ ✶✹✳✼ ✾✷✳✺ ✷✼✳✾ ✾✾✳✾

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✷✾ ✴ ✸✷

❙✐♠✉❧❛t✐♦♥ r❡s✉❧ts ❢♦r ♥♦♥ ❤✐❡r❛r❝❤✐❝❛❧ ♠♦❞❡❧s

✶ ▼♦❞❡❧ ✹ : Xt = ✵.✹Xt−✸ + ✵.✹Xt−✹ + ξt.

n ✶✵✵ ✺✵✵ ✶✵✵✵ ✷✵✵✵ log n √n

• κn

log n √n

• κn

log n √n

• κn

log n √n

• κn

❚ ✼✵✳✹ ✻✼✳✸ ✼✶✳✵ ✾✵ ✶✵✵ ✶✵✵ ✾✸✳✷ ✶✵✵ ✶✵✵ ✾✺✳✸ ✶✵✵ ✶✵✵ ❖ ✷✺✳✵ ✶✳✻ ✷✽✳✽ ✶✵ ✵ ✵ ✻✳✽ ✵ ✵ ✹✳✼ ✵ ✵ ❲ ✹✳✻ ✸✶✳✶ ✵✳✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸✵ ✴ ✸✷

◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❢♦r ❙P✺✵✵ ❧♦❣✲r❡t✉r♥s

▲♦❣✲r❡t✉r♥ ♦❢ ❙P✺✵✵ ❝❧♦s✐♥❣ ✈❛❧✉❡s ❢r♦♠ ✶✶✴✷✵✶✶ → ✶✶✴✷✵✶✻

❚❛❜❧❡ ✕ ❘❡s✉❧ts ♦❢ t❤❡ ♠♦❞❡❧ s❡❧❡❝t✐♦♥ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t ❛♥❛❧②s✐s ♦♥ ❋❚❙❊ ✐♥❞❡①✳

κn = log(n) κn = √n κn = κn

• m
• ❆❘❈❍(✶, ✶)
• ❆❘❈❍(✶, ✶)
• ❆❘❈❍(✶, ✶)
• Q✶✵(

m) ✾✳✸✵ ✾✳✸✵ ✾✳✸✵ p − value ✵✳✺✵ ✵✳✺✵ ✵✳✺✵

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸✶ ✴ ✸✷

❘❡❢❡r❡♥❝❡s

❇❛r❞❡t✱ ❑❛r❡ ❛♥❞ ❑❡♥❣♥❡ ✭✷✵✷✵✮✳ ❈♦♥s✐st❡♥t ♠♦❞❡❧ s❡❧❡❝t✐♦♥ ❝r✐t❡r✐❛ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ❢♦r ❝♦♠♠♦♥ t✐♠❡ s❡r✐❡s ♠♦❞❡❧s✳ ❊❧❡❝✳ ❏♦✉r♥✳ ❙t❛t✐st✳ ❇❛r❞❡t ❛♥❞ ❲✐♥t❡♥❜❡r❣❡r ✭✷✵✵✾✮✳ ❆s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t② ♦❢ t❤❡ ◗✉❛s✐✲▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❊st✐♠❛t♦r ❢♦r ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❝❛✉s❛❧ ♣r♦❝❡ss❡s✳ ❆♥♥✳ ❙t❛t✐st✳ ❉♦✉❦❤❛♥ ❛♥❞ ❲✐♥t❡♥❜❡r❣❡r ✭✷✵✵✽✮✳ ❲❡❛❦❧② ❞❡♣❡♥❞❡♥t ❝❤❛✐♥s ✇✐t❤ ✐♥✜♥✐t❡ ♠❡♠♦r②✳ ❙t♦❝❤✳ Pr♦❝✳ ❛♥❞ t❤❡✐r ❆♣♣❧✐✳ ❋r❛♥❝q ❛♥❞ ❩❛❦♦ï❛♥ ✭✷✵✶✵✮✳ ●❆❘❈❍ ▼♦❞❡❧s✳ ❲✐❧❡②✳ ❍s✉✱ ■♥❣ ❛♥❞ ❚♦♥❣✳ ❖♥ ♠♦❞❡❧ s❡❧❡❝t✐♦♥ ❢r♦♠ ❛ ✜♥✐t❡ ❢❛♠✐❧② ♦❢ ♣♦ss✐❜❧② ♠✐ss♣❡❝✐✜❡❞ t✐♠❡ s❡r✐❡s ♠♦❞❡❧s✳ ❆♥♥✳ ♦❢ ❙t❛t✐st✳ ▲✐ ❛♥❞ ▼❛❦ ✭✶✾✾✶✮✳ ❖♥ t❤❡ sq✉❛r❡❞ r❡s✐❞✉❛❧ ❛✉t♦❝♦rr❡❧❛t✐♦♥s ✐♥ ♥♦♥✲❧✐♥❡❛r t✐♠❡ s❡r✐❡s ✇✐t❤ ❝♦♥❞✐t✐♦♥❛❧ ❤❡t❡r♦s❦❡❞❛st✐❝✐t②✳ ❏♦✉r♥❛❧ ♦❢ ❚✐♠❡ ❙❡r✐❡s ❆♥❛❧②s✐s ❙✐♥ ❛♥❞ ❲❤✐t❡ ✭✶✾✾✻✮✳ ■♥❢♦r♠❛t✐♦♥ ❝r✐t❡r✐❛ ❢♦r s❡❧❡❝t✐♥❣ ♣♦ss✐❜❧② ♠✐ss♣❡❝✐✜❡❞ ♣❛r❛♠❡tr✐❝ ♠♦❞❡❧s✳ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠❡tr✐❝s✳ ❙tr❛✉♠❛♥♥ ✭✷✵✵✺✮✳ ❊st✐♠❛t✐♦♥ ✐♥ ❝♦♥❞✐t✐♦♥❛❧❧② ❤❡t❡r♦s❝❡❞❛st✐❝ t✐♠❡ s❡r✐❡s ♠♦❞❡❧s✳ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✳

❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸✷ ✴ ✸✷